Contents

What logic is

The science of valid inference from which we deduce or induce a conclusion based on premises

Argument - set of propositions that are called premises which are put forward as a reason to believe another proposition called the conclusion. This is called a Syllogism

I want to get to London by noon. It is necessary condition to catch the 10 o'clock train to get to London by noon. The conclusion is catch the 10 o'Clock train. Note that really the conclusion is the rearrangement of the premises but it does not really provide any insight but it is a sound argument. Drawing a conclusion is an inference. This is an an example of formal logic

Categorical proposition - has a subject and a predicate
Subject
All cows(subject) eat grass(predicate)

Four type of Categorical Propositions

i.All F is G

ii.No F is G

iii.Some F is G

iv. Some F is not G

Example for iii and iv . No cows have wings. Cows are mammals. Some mammals do not have wings. This type of Syllogism is called Barbara. Give example of celarant here

DeonticKant- the logic of moral discourse. Lying is wrong therefore don't lie. (Only one premise and a conclusion). Note, straight from the premise and then to the conclusion but without the desire. (i.e. I want to get to London by noon). Lying is wrong therefore there is no desire. If you want to add to the desire to do right implies the possible desire to do wrong and in that case you really do not understand what it is to do right. i.e. You do not steal something not because you would be found out - you do not do it because it is wrong.

Modal Logic

It is not possible for vixens to be female then that vixen is not female. If it is not possible it can not be actual

Logic of Conditionals

If that is gold then I'm a Dutchman????

Possible Worlds

Kripke. You might argue that if German won the war we would be speaking German but what you can say is that German did not win the war and we are not speaking German. He postulates possible worlds where German did win the war. This is looking at the limits of possibility. But there is there a no possible world where a circle is a square because it is part of the definition.

Deductive and Inductive Arguments

With deduction it gives you a certain conclusion/validity.

Valid
If it snows the mail will be late. It is snowing. Therefore the mail will be late

Invalid
If it snows the mail will be late. The mail is late therefore it is snowing. You should always be able to find an alternative reason.

With inducton it does not give you ascertainment but gives you probability. Everyday the Sun has risen therefore it will rise tomorrow. This is very, very probable but it is not certain.

Are the premises true so we can accept the conclusion - Deductive. You have a good argument
Are the premises true so that we have near certainly the the conclusion is valid - Deductive.

Circular argument. All whales are whales therefore all whales are whales. Correct but not a sound argument. Could you confusing people because people are always looking to valid arguments but someone can try can exploit people by putting in lots of other premises to hide the circular arguments. This is a form of informal logic.

Truth Tables

P

Q

P

(P-->Q)

Q

Where the first P and Q is the sentence letters. The P and P-->Q are the premises and the final Q is the conclusion

Arguments by AnalogyThe Blind Watchmaker. The universe is like a watchmaker therefore the universe has a maker. Not a good argument

London is in England and sugar is sweet, therefore Paris is the capital of France. Although both premises are true and conclusion is valid it is still not a good argument unless you can provide a context between the premises and the conclusion.

The Stoics developed a logic different from Aristotle's, and to a large extent independently from him.Whereas Aristotelian syllogistic is a term-logic, Stoic logic was propositional: it explored the relations between what they called "assertibles"—that is to say, sentences that can be used to make assertions. Assertibles can be simple ("It is day") or complex ("If it is day, it is light"/ "It is day or it is not light"). The argument forms classified by the Stoics involve one complex and one simple assertible: for example, "If it is day, it is light. It is day. So, it is light." This is the first of five "indemonstrables"—basic argument forms. The Stoics had a schematic way of representing the indemonstrables—what they called their "modes"—using ordinal numbers. The four remaining modes of the five indemonstrables are (2) If the first, then the second; not the second; so not the first; (3) Not both the first and the second; the first; so not the second; (4) Either the first or the second; the first; so not the second; (5) Either the first or the second; not the first; so the second. Since the assertibles could be either negative or positive, and the complex assertible could itself include complex assertibles ("If both the first and the second, then the third"), there was quite a wide range of indemonstrable argument schemes.

Medieval Logic

Medieval thinkers were troubled with Aristolian syllogism. Take this example.
I know she is in London or Manchester. Knowing that she is in either does not mean you know that she is in either and does not develop any form of knowledge.

Nineteenth Century Logic

George Boole wanted to show the laws of thought were rigorous as the laws of science. He notes algebraic link between and and multiplication and or and addition. One of his interpretation of this system was propositional logic where 1 = truth and 0 = false.

Leibnitz and Hobbes thinking is simply calculation. A logical language in which reasoning could be computer. He is inspired Frega and Jevons took Boolean idea and build a logic machine.

Frega took the following example
Jack (subject) Loves Jill (Predicate) and said that Jack and Jill can operate on the same operational level and have a two case predicate i.e. loves

Loves=Jack, Jill

He did this to formalize all these relations (symbolism) and show how inferences work. This came to be called logicism. Arithmetic could be reduced to logic and not rest on axioms or Empiricism and turned in to x, y and z with Boolean operators

Russell wrote to Frega explaining a contradiction in his system called the Russell Paradox.
e.g. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition.

For instance if the barber shaves all of the men who shave themselves in the village. Does he shave himself. If he does he doesn't. If he doesn't then he does. Russell solution was to not allow classes of the higher level to take members from the lower level but the lower cannot take members from the higher level.

Although Aristotle was criticized for trivial syllogism but the computer has taken many trivial arguments and compute them rapidly to perform thousand of operations.